# Generalised Jantzen filtration of Lie superalgebras I

@article{Su2010GeneralisedJF, title={Generalised Jantzen filtration of Lie superalgebras I}, author={Yucai Su and R. B. Zhang}, journal={arXiv: Representation Theory}, year={2010} }

A Jantzen type filtration for generalised Varma modules of Lie superalgebras is introduced. In the case of type I Lie superalgebras, it is shown that the generalised Jantzen filtration for any Kac module is the unique Loewy filtration, and the decomposition numbers of the layers of the filtration are determined by the coefficients of inverse Kazhdan-Lusztig polynomials. Furthermore, the length of the Jantzen filtration for any Kac module is determined explicitly in terms of the degree of…

## 15 Citations

Generalised Jantzen filtration of exceptional Lie superalgebras

- Mathematics
- 2016

Let g be an exceptional Lie superalgebra, and let p be the maximal parabolic subalgebra which contains the distinguished Borel subalgebra and has a purely even Levi subalgebra. For any parabolic…

Generalised Jantzen filtration of Lie superalgebras II: the exceptional cases

- Mathematics
- 2013

Let $g$ be an exceptional Lie superalgebra, and let $p$ be the maximal parabolic subalgebra which contains the distinguished Borel subalgebra and has a purely even Levi subalgebra. For any parabolic…

Jantzen filtration and strong linkage principle for modular Lie superalgebras

- MathematicsForum Mathematicum
- 2018

In this paper, we introduce super Weyl groups, their distinguished elements and properties for basic classical Lie superalgebras. Then we formulate Jantzen filtration for baby Verma modules in…

Jantzen Conjecture for singular characters

- Mathematics
- 2021

We show that the Jantzen filtration of a Verma module (possibly singular) coincides with its radical filtration. It implies that the Jantzen Conjecture on Verma modules holds for all infinitesimal…

The Kac–Wakimoto character formula for the general linear Lie superalgebra

- Mathematics
- 2015

Character formulas for Lie superalgebras have been shown to have important applications to number theory and combinatorics. We prove the Kac-Wakimoto character formula for the general linear Lie…

O ct 2 01 3 KAC-WAKIMOTO CHARACTER FORMULA FOR THE GENERAL LINEAR LIE SUPERALGEBRA

- Mathematics
- 2018

Character formulas for Lie superalgebras have been shown to have important applications to number theory and combinatorics. We prove the Kac-Wakimoto character formula for the general linear Lie…

Character and Dimension Formulae for Queer Lie Superalgebra

- Mathematics
- 2015

Closed formulae are constructed for the characters and dimensions of the finite dimensional simple modules of the queer Lie superalgebra $${\mathfrak{q}(n)}$$q(n). This is achieved by refining…

HIGHEST WEIGHT VECTORS OF MIXED TENSOR PRODUCTS OF GENERAL LINEAR LIE SUPERALGEBRAS

- Mathematics
- 2014

In this paper, a notion of cyclotomic (or level k) walled Brauer algebras ℬk, r, t is introduced for arbitrary positive integer k. It is proven that ℬk, r, t is free over a commutative ring with rank…

Mock modular index of M2-M5 brane systems

- Mathematics
- 2017

We present Bogomolny-Prasad-Sommerfield (BPS) indices of the supergroup Wess-Zumino-Witten
(WZW) models that live on intersecting M2-M5-brane systems. They can encode data of the stretched …

## References

SHOWING 1-10 OF 51 REFERENCES

Super duality and Kazhdan-Lusztig polynomials

- Mathematics
- 2004

We establish a direct connection between the representation theories of Lie algebras and Lie superalgebras (of type A) via Fock space reformulations of their Kazhdan-Lusztig theories. As a…

Composition factors of Kac modules for the general linear Lie superalgebras

- Mathematics
- 2004

The composition factors of Kac modules for the general linear Lie superalgebras are explicitly determined. In particular, a conjecture of Hughes, King and van der Jeugt in [J. Math. Phys. 33 (1992),…

Highest weight categories arising from Khovanov's diagram algebra III: category O

- Mathematics
- 2008

We prove that integral blocks of parabolic category O associated to the subalgebra gl(m) x gl(n) of gl(m+n) are Morita equivalent to quasi-hereditary covers of generalised Khovanov algebras. Although…

Filtrations on generalized Verma modules for Hermitian symmetric pairs.

- Mathematics
- 1988

The purpose of this paper is to analyze filtrations on generalized Verma modules for Hermitian Symmetrie pairs. We prove that the socle and radical series for these modules coincide and explicitly…

Cohomology of Lie superalgebras 𝔰𝔩m|n and 𝔬𝔰𝔭2|2n

- Mathematics
- 2004

We explicitly compute the first and second cohomology groups of the classical Lie superalgebras 𝔰𝔩m|n and 𝔬𝔰𝔬 2|2n with coefficients in the finite‐dimensional irreducible modules and the Kac…

On the composition factors of Kac modules for the Lie superalgebras sl(m/n)

- Mathematics
- 1992

In the classification of finite‐dimensional modules of Lie superalgebras, Kac distinguished between typical and atypical modules. Kac introduced an induced module, the so‐called Kac module V(Λ) with…

Highest Weight Categories Arising from Khovanov's Diagram Algebra I: Cellularity

- Mathematics
- 2011

This is the first of four articles studying some slight generalisations Hn m of Khovanov’s diagram algebra, as well as quasi-hereditary covers Kn m of these algebras in the sense of Rouquier, and…

Analogue of Kostant's u-cohomology formula for the general linear superalgebra

- Mathematics
- 2004

We determine in an explicit form the Lie superalgebra cohomology groups of the upper triangular odd subalgebra (Glm|n)+1 of the general linear superalgebra Glm|n with coefficients in irreducible…